Roman Domination in Complementary Prisms

The complementary prism GG of a graph G is formed from the disjoint union of G and its complement G by adding the edges of a perfect matching between the corresponding vertices of G and G. A Roman dominating function on a graph G = (V, E) is a labeling f: V (G) → {0, 1, 2} such that every vertex with label 0 is adjacent to a vertex with label 2. The Roman domination number γR(G) of G is the minimum f(V) = Σv∈V f(v) over all such functions of G. We study the Roman domination number of complementary prisms. Our main results show that γR(GG) takes on a limited number of values in terms of the domination number of GG and the Roman domination numbers of G and G